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The rational homotopy of Thom spaces and the smoothing of isolated singularities. (English) Zbl 0563.57010
Rational homotopy methods are used for studying the problem of the topological smoothing of complex algebraic isolated singularities. It is shown that one may always find a suitable covering which is smoothable. The problem of the topological smoothing (including the complex normal structure) for conical singularities is considered in the sequel. A connection is established between the existence of certain relations between the normal Chern degrees of a smooth projective variety and the question of its realization as a linear section (not necessarily hyperplane).
MSC:
57R10 Smoothing in differential topology
32Sxx Complex singularities
55P62 Rational homotopy theory
32S05 Local complex singularities
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[1] O. BURLET, Cobordismes de plongements et produits homotopiques, Comm. Math. Helv., 46 (1971), 277-288. · Zbl 0221.57017
[2] Y. FELIX, D. TANRE, Sur la formalité des applications, Publ. IRMA, Lille, 3-2 (1981).
[3] H. HAMM, On the vanishing of local homotopy groups for isolated singularities of complex spaces, Journal für die reine und ang. Math., 323 (1981), 172-176. · Zbl 0483.32007
[4] R. HARTSHORNE, Topological conditions for smoothing algebraic singularities, Topology, 13 (1974), 241-253. · Zbl 0288.14006
[5] R. HARTSHORNE, E. REES, E. THOMAS, Nonsmoothing of algebraic cycles on Grassmann varieties, BAMS, 80(5) (1974), 847-851. · Zbl 0289.14011
[6] S. HALPERIN, J.D. STASHEFF, Obstructions to homotopy equivalences, Adv. in Math., 32 (1979), 233-279. · Zbl 0408.55009
[7] M.L. LARSEN, On the topology of complex projective manifolds, Inv. Math., 19 (1973), 251-260. · Zbl 0255.32004
[8] J. MILNOR, Singular points of complex hypersurfaces, Princeton University Press, 1968. · Zbl 0184.48405
[9] S. PAPADIMA, The rational homotopy of thom spaces and the smoothing of homology classes, to appear Comm. Math. Helv. · Zbl 0592.57025
[10] E. REES, E. THOMAS, Cobordism obstructions to deforming isolated singularities, Math. Ann., 232 (1978), 33-53. · Zbl 0381.57011
[11] H. SHIGA, Notes on links of complex isolated singular points, Kodai Math. J., 3 (1980), 44-47. · Zbl 0442.57015
[12] A.J. SOMMESE, Non-smoothable varieties, Comm. Math. Helv., 54 (1979), 140-146. · Zbl 0394.14016
[13] D. SULLIVAN, Infinitesimal computations in topology, Publ. Math. IHES, 47 (1977), 269-331. · Zbl 0374.57002
[14] R. THOM, Quelques propriétés globales des variétés différentiable, Comm. Math. Helv., 28 (1954), 17-86. · Zbl 0057.15502
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